1. Stating the problem: Simplify the expression $\sqrt[3]{27} - \sqrt{3}$.\n\n2. Evaluate each root separately:\n- $\sqrt[3]{27}$ is the cube root of 27. Since $27 = 3^3$, we have $\sqrt[3]{27} = 3$.\n- $\sqrt{3}$ is the square root of 3 and cannot be simplified further.\n\n3. Substitute the evaluated values back into the expression:\n$$\sqrt[3]{27} - \sqrt{3} = 3 - \sqrt{3}.$$\n\n4. Final answer: The simplified form of the expression is $3 - \sqrt{3}$.